If $P$ and $Q$ are two sufficiently general points on a cubic surface, the line between them intersects the cubic surface at a unique third point, $f(P,Q)$. This gives a binary operation on (generic) points on a cubic surface.

For an elliptic curve, we can easily find the answer to the analogous question using the fact that $f(P,Q)=P^{-1}Q^{-1}$ in some abelian group. This also tells us the answer for identities with at most three variables - since any three points lie on a plane, the question reduces to the elliptic curve case. But what about four variables? Are there any identities with four or more variables that hold on a cubic surface but are not formally implied by the three variable identities?

Lastly, these identities clearly define a variety in the sense of universal algebra? What properties does this variety have? What does a free algebra on $n$ elements look like?